Tensor products of Leavitt path algebras

Abstract

We compute the Hochschild homology of Leavitt path algebras over a field $k$. As an application, we show that $L_2$ and $L_2\otimes L_2$ have different Hochschild homologies, and so they are not Morita equivalent; in particular they are not isomorphic. Similarly, $L_{\infty }$ and $L_{\infty }\otimes L_{\infty }$ are distinguished by their Hochschild homologies and so they are not Morita equivalent either. By contrast, we show that $K$-theory cannot distinguish these algebras; we have $K_{\ast }(L_2)=K_{\ast }(L_2\otimes L_2)=0$ and $K_{\ast }(L_{\infty })=K_{\ast }(L_{\infty }\otimes L_{\infty })=K_{\ast }(k)$.

10 pages. Added hypothesis to Corolary 4.5; Example 5.2 expanded, other cosmetic changes, including an e-mail address and some dashes. Final version, to appear in PAMS

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